The most famous quote about Complex Systems comes from Aristotle who said that "The whole is more than the sum of its parts".
Complex systems are systems where the collective behavior of their parts entails emergence of properties that can hardly, if not at all, be inferred from properties of the parts. Examples of complex systems include ant-hills, ants themselves, human economies, climate, nervous systems, cells and living things, including human beings, as well as modern energy or telecommunication infrastructures.
A good introduction to complexity is given by Murray Gell-Mann.
Complex Systems Science
THE WHOLE IS MORE THAN THE SUM OF ITS PARTS. - Aristotle
The new Science of Complex Systems is providing radical new ways of understanding the physical, biological, ecological, and social universe. The economic regions that lead this science and its engineering will dominate the twenty first century by their wealth and influence. In all domains, complex systems are studied through increasingly large quantities of data, stimulating revolutionary scientific breakthroughs. Also, many new and fundamental theoretical questions occur across the domains of physical and human science, making it essential to develop the new Science of Complex Systems in an interdisciplinary way.
This new science cuts across traditional scientific boundaries, creating new and shorter paths between scientists and accelerating the flow of scientific knowledge. Complex systems science bridges the natural and social sciences, enriching both, and reduces the gap between science, engineering, and policy.
It will also help reduce the gap between pure and applied science, establishing new foundations for the design, management and control of systems with levels of complexity exceeding the capacity of current approaches.
Funding this fundamental scientific research will be popular because its applications will impact on everyone’s life in many obvious ways including medicine, health, welfare, food, environment, transportation, web services. Thus Complex Systems Science will enhance long-term harmony between science and societal needs.